Optimal feedback control for constrained mechanical systems - Glaas, Daniel; Leyendecker, Sigrid (Proc. Appl. Math. Mech. (PAMM), 2016)
An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we want to use it to control constrained dynamical systems (differential algebraic equations of Index 3). To describe their discrete dynamics, a constrained variational integrators [1] is used. Using a discrete version of the Lagrange-d’Alembert principle yields a forced constrained discrete Euler-Lagrange equation in a position-momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (qopt, popt) and according control input uopt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input uR. Adding uopt and uR to the discrete Euler-Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinhei
Keywords: Variational integrator; feedback control; optimal control

Variational integrator based optimal feedback control for constrained mechanical systems - Glaas, Daniel; Leyendecker, Sigrid (ZAMM - Zeitschrift für angewandte Mathematik und Mechanik, 2018)
Today, a lot of mechanical systems have to operate with an improved performance compared to equal constructions decades ago. To stay competitive, engineers need to optimise all aspects of a mechanical system including its feedback control. An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller. To describe the discrete dynamics, a constrained variational integrator is used, which is a variant of a structure‐preserving integration scheme. The desired optimal trajectory and according control input is determined solving the discrete mechanics and optimal control (DMOC) algorithm based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equations yield the optimal deviation control input. Adding the optimal control input to the deviation control value causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. In this work, criteria for the choice of quadrature rules required to derive the variational integrator are investigated to close a gap of previous works in this field. This procedure is applied to three different coordinate choices, minimal, redundant and nullspace coordinates. Simulation examples show that a stable handling of highly‐nonlinear systems is assured. With applying the formulas derived in this work, a transformation between different coordinate parametrisations and the corresponding cost matrices leads to the same deviation control and thus to the same system behaviour. Nevertheless, it is in the responsibility of the application engineer to choose appropriate cost‐matrices for the occurring perturbation
Keywords: discrete Riccati equations; optimal control; variational integrators

Variational integrator for constrained mechanical systems with pulsed disturbances and optimal feedback control - Glaas, Daniel; Leyendecker, Sigrid (Proc. Appl. Math. Mech (PAMM), 2017)